function WrightFisher(N,p,total_generations)
current_gen=1
freqs=zeros(total_generations) #This is what we'll keep track of
freqs[current_gen] = p
while current_gen<total_generations
next = only(rand(Binomial(N,freqs[current_gen]),1))/N
freqs[current_gen+1] = next
current_gen += 1
end
return(freqs)
endLecture 1
Pre-amble
Welcome to the class
Welcome to Population Genetics, EEB5348!
Before we begin, some general logistics of the course.
The course will consist of a weekly lectures, paper/methods discussions and a lab.
- Trying out a bunch of new approaches - format might change if it feels like things are not working well.
Introduce yourselves!
Grading
Grades will consist of
- Participation in discussions. (24%)
- Weekly lab assignments/quizzes. (24%)
- Two minor analyses of a dataset. (25%)
- One final project. (25%)
And two free points!
Assignments/Project
The goal is to create a small set of scripts that you have for many basic population genetics tasks.
We will assume you have called variants somehow. We sadly don’t have time to teach you how to call variants on the many sequencing approaches one could take (see Jill Wegrzin’s EEB XXXX for that!).
You will develop scripts to calculate basic population genetic statistics, plot them, and perform statistical comparisons when possible in each weekly lab.
Two incremental code based submissions in which I’ll test your code on different data than used in class.
Project cntd.
Your term project will be to fully analyze the genetics of a population of your choice.
- Examine genetic diversity and its distribution
- Run basic demographic analysis
- Selection scans
- etc. . . .
You will meet with me half-way through the semester to decide on a tractable data-set, or I will provide you one
I’m suddenly worried I don’t have the background for this class
Don’t panic!
The goal of this course is to build your evolutionary intuition as well as knowledge of what different population genetic tools do. AI ramble goes here
Be comfortable asking questions!
Mini boot-camp on programming this week, math we’ll try to introduce as we go.
What language should I code in?
The nature of this course is to help you develop tools useful for you, and so no single language is being prescribed.
I will write code primarily in Julia, but pretty much anything here should be achievable easily in R or python, and with a bit more elbow grease in any language you desire.
Beyond that you will need at least a passing familiarity with bash, as some of the tools we’ll run are only really accessible via command-line.
Schedule
12 regular weeks, 2 project focused
You can find an updated schedule on the course website
Last two weeks - students’ choice!
Questions?
Population Genetics
What is population genetics?
Population genetics is the anvil on which evolutionary intuition is developed.
Many verbal models that sound reasonable work contrary to your ideas once spelled out.
Pop-gen models specifically consider how evolution actually proceeds and affects genetic variation.
What are models?
A model is a simplified version of reality.
In pop gen, we generally deal with three “types” of models:
Explanatory - how does this system work?
Predictive - what will happen in the future?
Statistical - how well is the data represented by a model?
The same model can really have both explanatory and predictive power (and be used as the backbone for a statistical model).
Why we model: a historical example
Imagine a population with two types of butterflies: red and white.
You, as a geneticist, work out the underlying control. You find the trait is controlled by a single gene, and is biallelic.
An allele is just a variant at some heritable site. It could be a single mutation, or a whole complex of co-inherited variants.
Biallelic sites are limited to two alleles.
You work out that AA and Aa butterflies are red, aa are white.
What will happen to the color in the long run?
Testing your intuition
Recall: Aa and AA are red, aa are white.
How do you compare to early 20th century geneticists?
It was broadly believed that dominant alleles should become more common.
The dominance itself was held as a selective advantage (the trait is “stronger”).
It took a mathematician to point out that there’s absolutely no reason dominance should have an advantage in inheritance of the allele.
Testing intuition with a model
Models begin by simplifying the world as much as possible. Given what we know about the system:
Single locus
Diploid
Sexually reproducing
Let’s get rid of other potential factors
Only dominance, no other fitness differences
No mutation
No migration/multiple populations
Population large enough we don’t need to worry about drift
Completely random mating
Modeling tips and tricks
It’s often useful to draw out a diagram of what your system looks like.
This lets you see what forces are relevant, what you can potentially simplify, etc.
Now let’s define what we are interested in
We want to know how the frequency of white butterflies \(P_{white}\) will change.
Let \(p\) be the frequency of the A allele. Then \(q=1-p\) is the frequency of a.
We also know that \(P_{white} = P_{AA}+P_{Aa}\)
What will \(P_{white}'\) be?
We often denote the next step/generation with a tick (`)
Time for chalkboard!
What we should have derived
\(P_{white}' = p*p+p*(1-p)+(1-p)*p= p^2+2pq\)
And additionally:
\(p' = P_{AA} + 1/2 P_{Aa} = p\)
So after one generation, you’ll always have \(P_{white}=p^2+2pq\)
You probably recognize this: Hardy-Weinberg
Originally, HW was derived to show that dominance does not lead to a bias in transmission, even if the trait is expressed over the alternate in heterozygotes.
Takeaways
Models have unexpected insights/uses. Hardy never imagined his short letter to Nature would become taught across schools in the world as a “null model” in evolution.
Null Models in Evolution
Throughout this semester, we’ll return to a few basic “null” models that will serve as useful measuring sticks.
Null models are often “zero force” models - they ask what happens when the interesting processes are not in play (“An object in motion stays in motion”).
One that we’ll come back to time and again because of its simplicity is the Wright-Fisher model.
Wright-Fisher
Sewall Wright, Ronald A. Fisher
Wright and Fisher were two of the fathers of population genetics.
They hated each other, and disagreed in most beliefs of how evolution actually acted.
They did agree, however, that one could describe what happens to evolution when there are no forces applied to a population.
The model
Population of constant size, N. N is one of two parameters of the model. Good models get a lot done with a few parameters.
Allele A starts at frequency \(p_0\). This is the only other parameter.
Diploid adults, haploid gametes.
Hermaphroditic.
Life cycle: gametes fuse to form adults, adults form gametes and die.
Random sampling - Binomial Distribution
The probability that \(k\) gametes carrying the A allele are selected for the next generation can be calculated using Binomial probability distribution:
\[ P(k|N,p) = \binom{N}{k}p^k(1-p)^{N-k} \]
Here, \(\binom{N}{k}\) counts the ways we could choose \(k\) gametes out of the \(N\) total.
Wright-Fisher In silico
Rather than start with math, let’s start with a simple simulation of the model:
What does that look like in practice?
What does it look like across many trials?
Wright-Fisher analytically
Let’s re-examine the model purely from a mathematical approach. We might want to know what the expected allele frequency in the next generation is. The expected value of a random outcome is the sum of all possible outcomes weighted by their probabilities.
\[ E[p_{t+1}] = \sum_{i=1}^N\left( \frac{i}{N} P(i,N|p_{t}) \right) = \frac{1}{N} E[Binom(N,p_{t})] \]
Where \(Binom(N,p)\) is the \(Binomial\) distribution, describing the chance of getting some number of successes given \(N\) trials with each having a chance of success of \(p\). If you know your distributions, you know that \(E[Binomial(N,p)] = Np\), so you get:
\[ E[p_{t+1}] = \frac{1}{N}Np_{t}=p_{t} \]
And, further, \(E[p_t] = p_{t-1} = ... = p_0\), meaning no change is expected.
Analytical vs Simulation
Looks like no real change, just as expected.
Variance in Wright-Fisher
The natural follow-up in probability is to describe the variance. In this case, the variance in the change from one generation to the next is
\[ Var[p_{t+1}] = \frac{p_t(1-p_t)}{2N} \]
But we might want to know what the value is after many generations. That is a bit more complex, but for now understand that it is proportional to the above result:
\[ Var[p_t] \propto \frac{t}{2N} \]
Add to the plot
Again, we can add this to the plot:
What is/are the important factors?
Our simple Wright-Fisher simulations and analytical results depend on two quantities:
The starting allele frequency.
The population size.
Let’s play around with each to develop a sense for what they do.
Allele frequency
Recall that expected allele frequency at time step t is just the initial frequency. But variance has a more complicated relationship:
Allele frequency in practice
In practice, you find the largest allele frequency swings at intermediate frequencies.
Effects of population size
Population size does not affect the expected allele frequency change.
Takeaway
While we generally say that drift is strongly affected by population size, it plays no role in the direction of drift. In the long run, small and large populations show the same trends in terms of drift.
Population size does affect the variance.
\(N\) vs Variance
No change in mean!
Different question: fate of an allele?
Let’s use this framework to now ask a different question: what’s the probability an alternate allele is fixed?
Let’s think of it formally - what is the probability that the A allele is fixed if it starts with frequency \(p_t\)?
Here, we can use a favorite trick of population geneticists: we’ll work backwards.
Imagine we started with a population where each of the N individuals has a different allele.
Eventually, once enough time has passed, all of the individuals in the population will have descended from just one of these ancestors.1
Who’s the ancestor?
If all of the alleles are neutral any of them have an equal probability of being the ancestor.
Now, what if there were two types of alleles in different proportions?
Should be fairly intuitive to see that:
\[ P(fix|p_0) = p_0 \]
Verifying with simulations
Here we start with a frequency of 0.2.
And if we start with a more common allele…
And the same happens with smaller populations
But what about real data?
Surprisingly, tracking individual allele frequency changes is not super common.
Either hard to get many generations (generation time ~year), or hard to catch every generation (multiple per day)
Often not the core interest of any study, but can be incedental to other interests.
Reading for Thursday - Leu et al. 2020 “Sex alters molecular evolution in diploid experimental populations of S. cerevisiae”
Plot of Data from Leu et al.
How general are the dynamics of this model? Here’s evolution in yeast:
How can we compare with WF expectations?
Do allele frequency changes seem odd in any way?
So, what happens to a new allele?
Homework Assignment (submit via HuskyCT):
What frequency will a mutation be found in when it first occurs?
What is the probability of that mutation becoming fixed?
Given only what you have learned here: should smaller populations show slower or faster rates of fixing substitutions?